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Dave Hewitt

School of Education, University of Birmingham, UK

Some throat clearing… in deciding what to write for the proceedings, I have decided to elaborate on some of the activities and themes we worked on within the session rather than report my interpretations of what actually took place. All activities I mention here did take place within the session (and in fact some additional activities also took place which I have decided not to comment upon here), however I have taken the liberty to expand on some of my thoughts which lay behind these activities, whereas in the session I offered little of this preferring to provoke thought in those present. End of throat clearing.

I am teaching… I see/hear a student do something relating to mathematics… I make a choice about how to respond (which may involve the choice not to say or do anything)…

What informs my decisions about how to respond? The nature of what was observed will be a factor, but what a student does is not the sole defining factor on the nature of a response. Sometimes I observe a teacherwho makes different choices, in response to something a student has said or written, compared with how I might imagine myself responding. Our beliefs about teaching and learning mathematics partly informs the way in which we respond as teachers, to such situations. As we continue educating ourselves our awareness of pedagogical situations develops and we begin responding in different ways to how we used to respond.

Through examining our responses to students we can begin to examine the awareness and beliefs which inform our decisions. I will start by considering a particular scenario and consider different forms of feedback which may be offered to someone in a learning situation. The situation does not involve explicit mathematical content (although I would argue that there is considerable geometry involved in the activity) however I suggest it can act as a metaphor when thinking about more explicitly mathematical situations.

Imagine a student blindfolded sitting at a table with a wastepaper bin somewhere the other side of the table.

Figure 1: a bird’s eye view of a blindfolded student sitting at a table with a wastepaper bin the other side of the table.

I will offer several scenarios relating to this situation and reflect upon teaching and learning issues for each in turn.

  1. The student is asked to throw a beanbag into the bin. Initially the student may complain because they do not know where the bin is.

After complaining, the student has a choice of whether to engage in this activity or not. If not, then the student opts out and may seek other things to occupy their time. If they decide to engage and throw the beanbag, they will have no awareness of where the bin is and so their attempt will not be a particularly informed one. The bin could be anywhere as far as they are concerned and so they may base the way they throw the beanbag on factors such as how comfortable the throw is on the throwing arm, or maybe they may imagine some throws they have done at other times in other contexts, such as sport, and try to replicate those. To an observer, it may appear as if the student has thrown the beanbag in a random manner.

At times students are asked to engage in some mathematics when they feel they have no idea how to proceed and would not necessarily spot when they have done something ‘correct’.Such times can lead to students opting out of the activity and finding other things to do which are not related to mathematics. So those students could become disruptive in the classroom and difficult for the teacher to manage. Even if the students decide to engage in the activity, they may not have relevant awareness which would informhow to proceed in an appropriate mathematical manner. They may appear to do things which seem completely random to a teacher. This is because their actions will be informed by something other than the mathematics which the teacher considers relevant. Without any further feedback on their actions, the students may just continue and have no sense of whether they are doing something relevant or whether they have or have not completed the original challenge.

  1. The blindfolded student throws the beanbag. Someone else in the room, whom I shall call Teacher 1, gives feedback to the student in the form of either Yes it is in the wastepaper bin or No it has missed the bin.

Now the student gets some feedback. However, the feedback only informs the student of whether the stated goal has been reached or not. If the beanbag missed the bin, there is no information within the feedback which might inform future throws, other than not to repeat the same throw. Even if the beanbag did go in the bin, the student will not have educated their awareness further for a similar activity in the future where the bin is re-positioned. The success gets attributed to luck rather than skill.

Sometimes mathematics work is marked with ticks and crosses. It is deemed to be either right or wrong. In this sense the feedback is similar to scenario B. However, the situation within mathematics might have different attributes. For example, with throwing the beanbag there are limitless possible ways of throwing the bag and where on the floor it might end up. It is this infinite set of possibilities which makes the yes/no response of such limited value. This is not always the case within some areas of the mathematics curriculum. For example, a student may be aware that a regular triangle is called either equilateral or isosceles but cannot recall which one. In this case a yes/no response might be very helpful. Thus, when there are limited options a yes/no response can be useful. However, it only is useful to inform a student to try an alternative option, not which option to choose. So there is little education of thestudent’s awareness which they use in selecting choices. This does not matter so much if it concerns a convention or a name (which I label as arbitrary, see Hewitt, 1999). In these situations, it is not a matter of educating awareness but of assisting memory. In a sense, a yes/no response provides a check for a student as to whether they have memorised successfully.

When properties or relationships are involved in an activity I suggest it is not just a matter of memorisation but it is about mathematical awareness (Hewitt, ibid). Suppose a student uses awareness appropriate to the task even though they arrived at an incorrect answer through a relatively small error. Being informed that the answer is wrong might spur the student to examine their work in a way where they can educate their own awareness. A yes/no response can be helpful for a student who is prepared and motivated to examine their own thinking with a level of detail which can be provoked knowing that something is wrong. It would not be so helpful for a student who used largely inappropriate mathematical awareness for the task.

  1. The blindfolded student throws the beanbag and another person, Teacher 2, offers feedback as to where the beanbag has landed with respect to the bin.

This type of feedback is informative in that it can be seen as providing non-judgemental information about where the beanbag has landed. It is significant whether the information is provided as to where the bin is with respect to the beanbag, or where the beanbag has landed with respect to the bin. In the former case, the feedback takes on the form of an instruction – for example, throw the beanbag more forward and to the left – which a student might then try to follow. In the latter case the student is left with needing to do some work to translate the feedback into how the next throw might be changed. With this subtle but significant difference they are told the consequences of their previous actions rather than being told how to act in the future.

An extension of this issue concerns aprinciple of whether a teacher believes that their role is to help a student use as little thought as possible to achieve an end result or whether they want a student to have to think in order to achieve an end result. Is it the result or is it the thinking which is important? If the former then the choice of activity and the choice of feedback is such that the level of challenge is small in order to maximise the chance of achieving the end result. If the latter then the choice of activity and the nature of feedback is such that the level of challenge is made high whilststill making the taskfeel achievable for a student using the awareness they possess. If the level of challenge is too high then a student may feel it to be impossible for them, and so give up thinking and start guessing instead.

  1. The blindfold is taken off the student, who can now observe where their beanbag lands.

Here the role of the teacher is in setting the activity and allowing a student to directly observe the consequences of their actions, rather than feeding back those consequences. The student can now see the goal – the bin – as well as seeing the consequence of a throw. This direct observation reveals more information than that which could have been provided in the previous scenario.The challenge of translating the visual observation into the physical action of muscles needs to be recognised. For example, I have recently observed my two-year old daughter, Tamsin, attempt to throw a model aeroplane down the hall. She throws and it goes up in the air above her head. She throws again, and it still goes above her head. She throws again and it lands behind her. Just because a student can observe directly the undesirable consequences of something they have done, it does not imply they know in what way they need to change what they did in order to achieve a desired result.

  1. Having observed the student miss the bin with several attempts, Teacher 3 invites the student to engage with some imagery. The teacher says imagine you have just thrown the ball into the bin successfully and a video was taken of this throw. Now play the video backwards so that the ball comes out of the bin and returns to your hand as your hand goes backwards towards your start position. Now physically act out the video being played forwards.

This form of feedbackis not a direct comment on the throw which has taken place but might be considered as an appropriate offering given the throws which have taken place. Thus, the teacher has observed some unsuccessful throws and considers what could be offered to the student which might help with the task. In this case it might initially appear to the student as if the teacher is offering something which is a distraction. There requires a willingness on behalf of the student to take their immediate attention away from their next throw and engage with the imagery being offered. This requires some will-power and it requires the student to have trust in the teacher that the potential payoff from what the teacher might offer is worth giving up the more immediate desire to continue throwing. How does a student gain that trust in their teacher? It seems to me that there may (or may not) be an initial sense of giving a new teacher the benefit of the doubt but, whatever the initial feeling, it is through the quality of the learning experiences a student has with a teacher that the student will decide whether to place trust in that teacher in the future.

In contrast to the beanbag and bin activity, I now offer a different activity. A collection of pentominoes and tetrominoes (Figure 2) are stuck on a board using blu-tack.

Figure 2: a collection of pentominoes and tetrominoes.

A student is asked to choose one of these shapes and place it on a different part of the board. I look at the shape and state whether I like that shape or not according to an unsaid rule that I am applying. If I like the shape it stays. If I don’t like the shape it goes in a reject pile. The task for the students as a whole is to try to bring over all the shapes I will like whilst trying to get as few rejections as possible. So, for example, after three shapes have been offered to me the situation might be as follows:

Accept / Reject

Figure 3: early stages of the activity.

Which shape would you choose to place before me next? The feedback I am offering is yes/no yet I suggest the feedback feels more helpful for the task than when teacher 1 offered yes/no for the beanbag and bin activity. In the pentominoes and tetrominoes activity there are times when a no can be just as useful as a yes. For example, if you are thinking of two possible rules which are consistent with my acceptance or rejection of the shapes in Figure 3, you might deliberately try a shape which fits in with one of your rules but not the other. Such a choice will mean that, whether I say yes or no, one of your rules will be discounted from then on.

The yes/no responses are sufficient to provide much mathematical activity in finding properties of shapes, thinking about logic in terms of if… then statements, and also considering issues such as how you ever know for sure whether a rule you are considering is the ‘same’ as the rule I am using.So yes/no feedback can help generate considerable thought in the shapes task, whereas it did little for the beanbag and bin task. The choice we make in the nature of our feedback is concerned with our awareness of mathematics as well as our awareness and beliefs about pedagogy.

I will finish by offeringthe following scenario:

As part of a small group activity, you observe two students trying to decide which was the lowest out of the decimal numbers 0.853 and 0.358. One student says 0.853 is the lowest, the other thinks 0.358 is the lowest.

As a teacher, what might you do in such a situation?

I offer below six different responses which you might make as a teacher and invite you to consider one at a time and catch your feelings about each suggestion. Once you have caught your feelings, work a little harder to account for why you have those feelings. I suggest that such examination can help you articulate some of your beliefs and frameworks which inform the way in which you work with students. I strongly suggest that you work seriously on one response before even reading the next response. To help discipline yourself I recommend you get a sheet of paper where you can cover up all the responses except the one which you are currently working on (and, of course, those that you have already worked on). To help a little with your discipline I have started the scenarios on a new page. So do not read further until you have got your piece of paper and covered up the following page.

Scenario 1

Ask one student to explain to the other why they thought their decimal was the lowest and vice versa.

Scenario 2

Invite them to do two subtractions:

0.853 – 0.358 –

0.358 and 0.853

___

Scenario 3

Explain what each of the digits are ‘worth’.

Scenario 4

Ask them what each of the digits are ‘worth’.

Scenario 5

Comment that they both cannot be right (and wait to see whether they begin to discuss the maths involved).

Scenario 6

Write both decimals on the board, one under the other, and reveal columns one number at a time from left to right:

0.

0.

0.8

0.3

0.85

0.35

0.853

0.358

Reference:

Hewitt, D. (1999), 'Arbitrary and Necessary: Part 1 a Way of Viewing the Mathematics Curriculum', in For the Learning of Mathematics, 19(3), 2-9.